-
The present invention relates to the field of biology
and computer science, in particular it relates to the use of
biological molecules for computational purposes.
-
Biological molecules such as nucleic acid and protein
are complex polymers of rather simple molecules. DNA
(deoxyribonucleic acid) is an unbranched polymer used by
organisms to store their genetic information. A DNA polymer
is usually referred to as a DNA strand and is composed of
monomer molecules, which are called nucleotides. Each
nucleotide is connected to the next in a polymerization
process. Nucleotides differ in their bases, of which some
typical representatives are: adenine (A), guanine (G),
thymidine (T) or cytosine (C). Considering that for each
position in a DNA strand at least four possible bases are
possible it is not difficult to imagine that many different
sequences can be generated. In fact with each added
nucleotide the number of possible combinations can be
increased by a factor of four.
-
So, a DNA strand comprises a sequence; this is the
sequence of the nucleotides from one end of the DNA strand to
the other. The ends of a DNA strand are chemically different:
one end is called the 5' end while the other end is the 3'
end. So single-stranded DNA has a sequence and an
orientation.
-
There are several features of DNA molecules which makes
them in principle attractive for computing purposes, we name
here three of them: (1) Watson-Crick complementarity (2) the
availability of natural enzymes which can recognize DNA
sequences and (3) the potential for massive parallelism.
-
The use of computing with biological molecules allows
for the solving problems which presently do not have feasible
(in time) solutions within the available silicon based
technology. Computing with biological molecules and in
particular DNA molecules, can solve many of such problems,
because the massive parallelism of DNA strands allows the
trillions of operations taking place simultaneously.
-
A natural target class of problems which require massive
parallelism are the so-called NP complete problems (see for
a description: M. R. Garey and D. S. Johnson, Computers and
Intractability, A Guide to the Theory of NP-Completeness,
W.H. Freeman and Co., San Francisco, 1979.). Perhaps the most
famous of the problems from this class is the satisfiability
(SAT) problem for Boolean formulas (explained below).
Adleman (L. M. Adleman, Science, 266: 1021-1024, 1994) was
the first to conduct an experiment that constituted a "proof
of principle" for the use of DNA computing for solving an NP-complete
problem. After this pioneering work a number of DNA-based
computing methods have been investigated. The method
used by Adleman is the filtering method. It starts with a set
of DNA molecules which represent all possible assignments to
all variables of a given problem (called the combinatorial
library) and then filters out the molecules corresponding to
good solutions. Lipton (R. J. Lipton, Science, 268: 542-545,
1995) has outlined a solution for the SAT problem using the
filtering method.
-
The current methods for computing with biological
molecules have, as outlined above, in principle an enormous
advantage over the silicon based technologies.
Technologically, however, the current methods for computing
with biological molecules are cumbersome. Most methods
require some knowledge of characteristics of the desired
solution in order to filter out the desired solution.
-
The present invention provides a method for finding a
potential solution to a computational problem though a DNA
computing method. In this aspect of the invention, the method
utilizes a library of biological molecules. The library
comprises a number of biological molecules that represent a
set of combinations of values for variables of the
computational problem. Upon generation of the library, it is
not known whether a specific instance of the computational
problem is solvable, i.e. has a solution. To determine
whether the problem instance is solvable one can determine
whether the library comprises a biological molecule that
represents a solution to the problem instance. In the art,
many methods have been proposed to find such a solution, all
with their particular advantages and disadvantages. In the
present invention we provide a method that is based on
preventing detection of combinations of values for variables
to said computational problem that do not represent a
solution. Prevention of detection is achieved by blocking at
least one biological molecule that represents a combination
of values for variables to said computational problem that do
not represent a solution. Blocking is achieved by providing
the library with a blocking agent.
-
The method of the invention is very versatile in that
many different computational problems can be solved. It is
often much easier to find a combination of values that do not
represent a solution (i.e. represent a false solution) than
it is to find a combination of values that represent a
solution to the computational problem (also referred to as a
true solution). In fact, for many computational problems it
is possible to easily determine all possible false solutions
to the computational problem. Many computational problems can
be expressed as boolean formulas. Boolean formula's can be
written into a conjunctive normal form (CNF). The conjunctive
normal form representation of a computational problem allows
a person skilled in the art to determine rapidly all
combinations of values for variables that represent false
solutions to the problem. This process can also easily be
automated using a computer. In the present invention it is
possible to generate one or more blocking agents that are
capable of blocking essentially all biological molecules that
represent false solutions to the computational problem,
leaving essentially only true solutions unblocked (if any
exist). In addition, if one solution is found in the library,
a blocking agent capable of blocking this solution may be
provided to the library, thereby leaving only other solutions
to the problem unblocked, (if they exist). This process can
be repeated until essentially all solutions to the problem
have been found. A particular advantage of the present
invention over the current filtering methods is that it can
be performed in essentially one step. By blocking detection
of essentially all biological molecules representing
combinations of values that represent a false solution to the
problem one can in one step detect whether the problem
comprises a true solution. Repeated incubations with one or
more blocking agents and selections of part of the biological
molecules are possible in the present invention, but not
required.
-
Although, preferably all biological molecules
representing false solutions are blocked by a blocking agent,
the present invention is already useful when a limited number
of false solutions are blocked. Partial blocking at least in
part limits the search for a true solution using a method of
the art. A method of the invention can therefore easily be
combined with a method in the art, for instance a filtering
method, to simplify the search for a solution.
Thus, in one aspect the invention provides the use of a
method for determining whether a specific biological molecule
is present in a library of biological molecules, wherein said
library represents a set of combinations of values for
variables of a computational problem, the method comprising
providing said library with a molecule capable of associating
with at least one biological molecule representing a
combination of values for variables to said problem, wherein
said association marks said at least one biological molecule
as a combination of values representing a false solution to
said problem, and determining whether said specific
biological molecule is present in said library. The marking
of a combination of values that represents a false solution
(if it exists)leaves a combination that represents a true
solution unmarked. By providing the library with sufficient
molecules to associate with essentially all false solutions
to said computational problem, only those biological
molecules that represent true solutions to the problem are
left unmarked. This, of course, under the assumption that the
considered problem instance is solvable, i.e. comprises a
true solution. When a biological molecule representing a true
solution is detected, it may be identified. Identification
entails the determination of the specific combination of
values represented by said biological molecule. Upon
identification of a true solution a blocking agent can be
devised capable of blocking detection of biological molecules
representing this solution in the library. The library can be
provided with this blocking agent. A method of the invention
can then be used to determine whether said library comprises
another true solution to said problem. Thus. in one
embodiment the invention provides a method of the invention,
further comprising blocking an identified biological molecule
representing a true solution of said problem and identifying
in said library, a possibly present biological molecule
representing a combination of values for variables, which
combination is a another true solution to said problem.
-
The marking of biological molecules can be used to
discriminate between marked and/or unmarked biological
molecules and thereby between biological molecules
representing respectively false or true solutions to said
computational problem. In a preferred embodiment blocking of
detection of a marked biological molecule is achieved using a
blocking agent capable of blocking detection of said at least
one biological molecule. Blocking of detection of a
biological molecule representing a false solution to said
computational problem at least limits the search for a
biological molecule representing a true solution to said
problem. In a preferred embodiment the detection of
essentially all biological molecules representing a false
solution to said computational problem is blocked. Thus if a
biological molecule is detected this means that the problem
comprises at least one solution. A blocking agent is a
physical and/or any other means for enabling elimination of
detection of a biological molecule.
-
Non-limiting examples of biological molecules that can
be used to generate a library of molecules that represent a
set of combinations of values for variables of a
computational problem comprise nucleic acid, protein and/or
lipid, or a functional equivalent of these biological
molecules. Preferably, said biological molecule comprises
nucleic acid. A functional equivalent of nucleic acid
comprises the same base-pairing capabilities as nucleic acid
in kind not necessarily in amount.
-
In a preferred embodiment said at least one
combination of values for variables of said computational
problem comprises a false solution to said computational
problem.
-
Detection of unblocked biological molecules is
preferably performed with an amplification step. In this way
the number of unblocked molecules is increased relatively to
the number of the blocked molecules thereby making the
detection a lot easier. Thus, in one embodiment the use of
the invention further comprises subjecting said library to an
amplification step, wherein said blocking agent is capable of
at least in part preventing amplification of said at least
one biological molecule representing at least one combination
of values for variables of said computational problem.
-
However, detection of unblocked molecules does not
have to be performed in an amplification step. It is very
well possible to devise other methods of detecting unblocked
molecules. For instance, one can combine the biological
molecules with a fluorochrome that upon blocking with a
blocking agent becomes quenched. In this way detection of
unblocked molecules can be done by simply determining whether
fluorescence can still be detected.
-
Another way of detecting an unblocked biological
molecule comprises providing a digestion signal to the
biological molecule with the blocking agent. The presence of
undigested biological molecules in the library then reflects
the presence of unblocked molecules.
-
Although unblocked molecules can be detected in many
ways, biological molecules are preferably detected using an
amplification step. The amplification step does not have to
be performed for detection of unblocked molecules. It can
also be performed to increase the amount of molecules to
facilitate further handling of the material. Preferably, the
amplification step comprises a nucleic acid amplification
reaction such as polymerase chain reaction and/or a nucleic
acid amplification in a cell.
-
A computational problem often can be represented
through two or more sub problems. This feature can be used to
design and/or choose the blocking agent. Defining sub
problems to a computational problems is advantageous for the
present invention. Particularly when for a desired solution
the outcome of all sub problems must be true, it may be
easier to design the blocking agent. In this case, finding an
agent capable of blocking a part in a biological molecule
representing a false solution of a sub problem identifies the
biological molecule as having not the desired solution of the
complete problem. In a preferred embodiment of the invention
said at least one blocking agent is capable of blocking at
least a part of said at least one biological molecule wherein
said part represents at least one combination of values for
variables of a sub problem of said computational problem. An
additional advantage of defining sub problems is that all
biological molecules comprising a representation of a
particular false solution of a sub problem can be blocked by
the same kind of blocking agent, irrespective of the
particular representations of solutions of other sub problems
in the biological molecules. Considering that in a preferred
embodiment of the invention all false solutions of the
computational problem are blocked by a blocking agent it is
in this case entirely possible that more than one blocking
agent is blocking a particular biological molecule.
-
Many computational problems can be represented through
Boolean formulas. The Boolean formulas are particularly
suitable for encoding solutions of a computational problem
into a library of biological molecules. A Boolean formula can
be given a Conjunctive Normal Form (CNF) from. It consists of
a set of clauses linked together by the conjunction "and"
operator. Therefore if an assignment for all variables is a
solution of a CNF formula, then each and every clause in the
formula must be true. Thus a clause can be seen as a sub
problem of the complete problem. The computional problem has
a solution only when all of the clauses of the CNF
representation of the Boolean formula are true. Therefore, in
a preferred embodiment a blocking agent is capable of
blocking a part of a biological molecule in said library that
represent a false solution of at least one clause of said CNF
representation of the mathematical problem. Preferably, said
CNF representation is a 3 CNF representation.
-
A library of biological molecules can be generated
from a number of different biological molecules provided that
it comprises sufficient information storage capacity.
Examples of suitable biological molecules are nucleic acid
derived molecules, protein and lipids. Preferably, said
biological molecule and/or said blocking agent comprises
nucleic acid or a functional analogue thereof.
-
In one aspect of the invention the basic principle of
blocking is based on the fundamental Watson-Crick
complementarity property of DNA. This means that, due to
their chemical nature, two DNA strands can become bonded
resulting in a helical double-stranded DNA molecule, the
famous double helix (Watson and Crick, 1953). Bonding of DNA
strands arises from the specific pairing (formation of
hydrogen bonds) of the bases: adenine (A) always pairs with
thymidine (T) and guanine (G) always with cytosine (C). The
complementary strands of a double-stranded molecule are
arranged in an anti-parallel fashion. This means that the two
stands are in a 'head-to-tail' arrangement: the 5' to 3'
orientation of one strand corresponds to the 3' to 5'
orientation of the complementary strand. Raising the
temperature can lead to separation of strands of a double-stranded
DNA molecule resulting in two single-stranded DNA
molecules. This process is called melting. If after melting
the temperature is slowly lowered, the complementary strands
will anneal to form the original double-stranded helical
molecule again. When a short oligonucleotide (called primer)
is annealed to a single stranded DNA molecule, this
oligonucleotide can serve as a primer for an enzyme, called
DNA polymerase, to produce a second strand of complementary
DNA.
-
There are several technical approaches that can be
followed for the inactivation of DNA molecules and subsequent
detection. One method which we describe in detail is
inactivation for replication using peptide nucleic acid (PNA)
blocking, and PCR (Polymerase Chain Reaction) detection. As
an alternative to PCR, of course, any many for nucleic acid
amplification can be used. PCR is a technique used to amplify
specific DNA strands in vitro. For PCR the nucleotide
sequence of the ends of the DNA strands to be amplified has
to be known. This is necessary because short oligonucleotides
primers complementary to the end of the DNA strands to be
amplified have to be synthesized. A PCR 'cycle' consists of:
(1) melting of the double-stranded target DNA resulting in
single-stranded target DNA molecules, (2) cooling to allow
annealing of specific primers to the target DNA, and (3)
extension of the primers by the enzymatic activity of DNA
polymerase. It is very important to realize that the
extension products of one primer can serve as a template for
the other primer in the next cycle, so each cycle
(theoretically) doubles the content of target DNA. The DNA
polymerases commonly used for PCR are thermostable, so they
retain activity despite the high temperatures during the
melting periods. In this example, PCR amplification of faulty
solutions is blocked, so that only the desired solutions are
amplified. This blocking can be achieved through the addition
of specific small peptide nucleic acid (PNA) molecules to the
PCR reaction mixture. PNA molecules are single-stranded DNA
mimics with a pseudopeptide backbone (see for a detailed
description M. Egholm et al., Nature 365:566-568, 1993).
PNA's are functional equivalents of nucleic acid. PNA's have
been shown to hybridize sequence-selectively to complementary
sequences of DNA, forming Watson-Crick double helices.
Moreover, PNA's do so with higher affinity than comparable
DNA molecules. So, by adding PNA molecules that anneal
specifically to the target DNA molecules in the same region
as the DNA primers, the latter cannot anneal. If the target
molecules represent faulty solutions, they will have PNA's
annealed to them instead of DNA primers. Because a PNA cannot
serve as a primer for DNA polymerase, polymerization, and
hence amplification of the target DNA, is prevented.
Therefore in one embodiment of the use of the invention said
blocking agent comprises peptide nucleic acid.
-
After PCR the amplified DNA molecules, representing
the good solutions, can be separated and visualized using
many known methods including DNA-chip technology. We also
want to mention some alternatives for blocking/detection of
DNA molecules. One can also block DNA molecules by making
them not accessible for restriction enzymes. This can be done
by adding PNA oligonucleotides which anneal to the target DNA
molecule at the position of a recognition site of a
restriction enzyme. The detection reaction is then based on
cloning the restricted molecules in a plasmid vector which is
replicated in vivo. Alternatively, the detection can be
performed on the basis of size of a DNA molecule which is
given in terms of the number of nucleotide base-pairs per
molecule. In this embodiment of the invention solutions
exist, if molecules smaller than the standard size exist in
the reaction buffer.
-
The rapidly developing technique of DNA-chip
technology provides a helpful tool for the readout of the
solutions. It is preferred that the library is present on the
chip and the blocking agent is added to the chip. Preferably
the biological molecules are arranged such that individual
molecules are in physically separated positions of the chip,
for instance in an array format. Unblocked biological
molecules can then be identified by finding the positions not
comprising the blocking agent. On the other hand, the chip
may comprise (an array of) blocking agents capable of
blocking a variety of biological molecules in the library.
The library can then be added to the array. Unblocked
molecules can then be detected in the fraction not associated
with the blocking agent.
-
A person skilled in the art is able to generate other
arrangements of the biological molecules and/or blocking
agents on the chip such that the number of different
biological molecules and/or blocking agents per position is
more than one, while still being able to detect unblocked
molecules. Reference is made to a previously mentioned
example using a fluorochrome.
-
Considering the chip implementation of the use of the
invention, a preferred embodiment of the invention comprises
the use of the invention wherein said library of biological
molecules and/or said blocking agent is physically linked to
a solid surface. A solid surface is advantageous for many
purposes not in the least for detection purposes and handling
purposes. The mentioned chip format is desirable when either
the library or the blocking agent is present in a
multiplicity of compartments and wherein each of compartment
comprises one type of biological molecule and/or one type of
blocking agent. The compartments may also comprise more than
one type of biological molecule or blocking agent.
-
Preferably, the blocking agent and/or the biological
molecule comprises a label, preferably a fluorophore such as
a fluorescent label. The fluorophore allows discrimination
between marked and unmarked biological molecules. A
fluorophore preferably comprises a fluorescent label. In a
non-limiting example of this embodiment of the invention we
describe the use of a blocking agent comprising a label in
combination with a library of biological molecules present in
an array on a chip. Providing the library with one labeled
blocking agent will identify a position in the array
comprising a biological molecule representing a false
solution the problem. When each position comprises
essentially only a biological molecule representing one
combination of values then one can easily find biological
molecules representing a true solution to the problem by
providing the library with labeled blocking agents capable of
associating with essentially all biological molecules
representing false solutions to said problem. In this case
positions that are left unlabeled comprise a true solution to
the problem. Detection of positions not containing a label
thus identifies such true solutions. In another preferred
embodiment of the invention, quenching of fluorescence is
used to discriminate between marked and/or unmarked
biological molecules. When all elements of the combinatorial
library have been labeled by a fluorescent dye the elements
which do not represent a solution can in a preferred
embodiment, be blocked for fluorescence by oligonucleotides
(or PNAs) which, through annealing, inactivate, or at least
decrease in a detectable way, the fluorescence of the target
DNA molecule. This inactivation can be achieved by the well-known
principle of "quenching". Quenching of fluorescence can
result from the presence of another fluorescent dye at the
blocking oligonucleotide (or PNA), which by annealing to the
target nucleotide, comes in close vicinity of the fluorescent
dye attached to the biological molecule. As a non-limiting
practical methodology to apply this principle we propose to
link the fluorescently labeled combinatorial library to the
surface of a chip. After annealing with the fluorescently
labeled blockers, one can read out the solutions by detection
of the positions of the chip which remain fluorescent. Such a
chip, based on the blocking of fluorescence can be described
as a "quenching chip readout". We also want to mention that
this methodology can be easily combined with other DNA-based
computing methods such as for example filtering.
-
In another aspect the invention provides the use of a
blocking agent for disabling detection of a biological
molecule, representing a combination of values for variables
of a computational problem, in a library of biological
molecules, wherein said library represents a set of
combinations of values for variables of said computational
problem. In another embodiment the invention provides the use
of a blocking agent for enabling elimination of detection of
a biological molecule in a library of biological molecules
representing a set of combinations of values for variables of
a computational problem.
-
In a preferred embodiment the invention provides a
method or a use of the invention, wherein said computational
problem comprises a SAT problem and/or a SAT related problem.
Examples
-
We illustrate the use of the blocking method for the
satisfiability (SAT) problem using the PCR method only. Let V
= { p1, ...,pn} be a set of Boolean variables -- their values
may be only 0 and 1 (0 stands for "false" and 1 stands for
"true"). A literal is either a variable pi or its negation
¬pi, we say that pi, ¬ pi are literals for pi.
-
We consider two logical operations: v ("or") and Λ ("and"). A
clause E is an expression of the form t1 v ... v tm where
each ti is a literal; for the purpose of this example we may
assume that for each variable pi there is at most one literal
for pi in E. A Boolean formula (in conjunctive normal form,
CNF) is an expression of the form E1 Λ ... Λ Em where each Ei
is a clause.
-
An assignment is a function on V which for each pi has the
value either 0 or 1. To compute the value of a literal (for a
given assignment ) we use the rule: ¬ 0 = 1 and ¬ 1 = 0. To
compute the value of a clause we use the rule: 0 v 0 = 0 and
0 ∨ 1 = 1 ∨ 0 = 1 ∨ 1 = 1. To compute the value of a formula
we use the rule: 0 ∧ 0 = 0 ∧ 1 = 1 ∧ 0 = 0 and 1 ∧ 1 = 1.
-
We say that an assignment satisfies a formula Φ if the
value (Φ) of Φ under is 1. Otherwise falsifies Φ (and
is a falsifier of Φ). We say that Φ is satisfiable if there
is an assignment satisfying Φ.
-
Example: Let V = {p1,p2,p3} be a set of variables and let Φ =
E1 ∧ E2 ∧ E3 be the Boolean formula over V such that E1 = p1 ∨
¬ p2, E2 = p1 ∨ p2 ∨ ¬ p3 and E3 = ¬ p1 ∨ ¬ p3.
-
Let 1 be the following assignment: p1 = 0, p2 = 1, p3 = 0.
Then 1(E1) = 0 ∨ 0 = 0, 1 (E2) = 0 ∨ 1 ∨ 1 = 1 and 1(E3) = 1
∨ 1 = 1. Thus 1 (Φ) = 0 ∧ 1 ∧ 1 = 0. Let 2 be the
assignment: p1 = 0, p2 = 0, p3 = 0. Then 2(E1) 0 ∨ 1 = 1,
(E2) = 0 ∨ 0 ∨ 1 = 1, and 2(E3) = 1 ∨ 1 = 1. Thus 2(Φ) = 1
Λ 1 Λ 1 = 1. Hence 1 falsifies Φ, 2 satisfies Φ, and Φ is
satisfiable.
-
The Satisfiability Problem (SAT) is to determine whether or
not an arbitrary Boolean formula Φ is satisfiable. Note that
SAT does not require that one finds a satisfying assignment
in the case that Φ is satisfiable.
-
The Find Satisfiability Problem (FIND SAT) is to determine
whether or not an arbitrary Boolean formula Φ is
satisfiable, and if it is then to give an assignment
satisfying Φ. In the sequel we assume that we have an
infinite sequence of (available) variables p1,p2,p3,... and
whenever we consider the case of n variables, the variables
are: p1,p2,...,pn.
Blockers
-
To start with, we need to code for each (Boolean) variable pi
its two possible values pi = 1 and pi = 0. Let qi (1) be a
single strand coding the value p1 = 1 and qi (0) be a single
strand coding the value pi = 0.
-
Example: The coding qi (1) = A and qi (0) = C for all 1 ≤ i ≤ n is
independent of a variable - the value 1 is always coded by A
and the value 0 is always coded by C.
-
Now we can code all possible assignments of variables by
single strands. To this aim, for a given number of variables
n, a n-strand is a strand of the form f1f2...fn where each fi
is either qi (1) or qi (0). The set of all n-strands is denoted by
Sn.
-
For a n-strand s, we use asg(s) to denote the corresponding
assignment , and for an assignment we use str() to denote
the corresponding n-strand. A blocker of a n-strand s is its
complement, it is denoted by b(s). Now, given a Boolean
formula Φ = E1 ∧ E2 ∧... ∧ Em over n variables, for each
clause Ei a blocker of Ei is a blocker of a n-strand s such
that asg(s) is a falsifier of Ei. The set of all blockers for
Ei is denoted by B(Ei). Then the set of all blockers for Φ,
B (Φ) is the union of B(Ei) for all clauses Ei of Φ; thus B(Φ)
= B(E1) ∪ ... ∪ B(Em).
-
For example, let n = 3 and let Φ = E1 Λ E2 where E1 = ¬ p1 ∨
p3 and E2 = ¬ p1 ∨ ¬ p2 ∨ ¬ p3. The falsifiers for E1 are 1
and 2 where 1(p1) = 1, 1(p2) = 1(p3) = 0, and 2(p1) = 2(p2)
= 1, 2(p3) = 0. The falsifier for E2 is 3 such that 3(p1) =
3(p2) = 3(p3) = 1.
-
Hence if we use the coding from Example 1 then the blockers
for E1 are TGG and TTG, because str (1) = ACC and str(2) =
AAC. The blocker for E2 is TTT becuase str(3) = AAA. Hence
the set of blockers for Φ is B(Φ) = { TGG, TTG, TTT}.
An Algorithm for SAT
-
We begin with an initial solution Z
0 that contains the set S
n
of all n-strands. To know S
n, we need to know only the number
of variables n (without knowing Φ). Thus we assume that such
a solution is prepared in advance -- it is a "ready product
on a shelf". This idea is common to filtering methods. Here
is an algorithm (
aaaaaa 1) for solving SAT.
ALGORITHM
aaaaaa 1
Input: A Boolean formula Φ of n variables.
- 1. Add B (Φ).
- 2. PCR.
- 3. PCR Successful?
If so, go to 5.
If not, go to 4. - 4. Output "NO" and Stop.
- 5. Output "YES".
- 6. Stop.
-
-
Once we know the input formula Φ, we proceed to Step 1 and
add B (Φ) to Z0 obtaining Z1. The intention of this step is to
"block" (by annealing) all the n-strands which represent
assignments that falsify Φ.
-
In Step 2, Z1 is PCR'ed and Z2 is obtained. Here the only n-strands
that can be successfully multiplied by PCR are the n-strands
that have not been blocked in Step 1 (after the
blockers from B(Φ) have been added). But these are precisely
the n-strands s such that the assignment asg(s) satisfies Φ.
Thus the PCR here is successful if and only if there exists
an assignment satisfying Φ.
-
In Step 3 we check whether or not the PCR from Step 2 was
successful. This is the case if the volume of Z2 is "clearly"
larger than the volume of Z1.
-
If the PCR was not successful, then we proceed to Step 4,
print "NO", and stop.
-
If the PCR was successful, then we proceed to Step 5, print
"YES", and stop in Step 6.
-
It must be clear by now that this algorithm prints "YES" (and
stops) if and only if Φ is satisfiable.
-
We continue our example: here n = 3 and S3 = { CCC, CCA, CAC,
CAA, ACC, ACA, AAC, AAA }. Since B(Φ) will anneal to their
complements in S3, the set of single strands in Z1, ss (Z1),
equals S3 - B(Φ). Hence ss(Z1) = { CCC, CCA, CAC, CAA, ACA }.
It is easily seen that indeed asg(ss(Z1)) is the set of all
assignments satisfying Φ. Since this set is not empty, the
PCR from Step 2 will be successful, and so the algorithm will
output "YES" and stop.
-
We feel that the following comments concerning the above
algorithm are needed here, even before we discuss later the
"laboratory implementation" of the algorithm.
- (1) We may construct B(Φ) by reading Φ from left to right,
clause by clause, as follows. Let E be a clause of Φ, and we
assume that literals in E are ordered according to the order
p1,...,pn of variables. Let, e.g., E = p1 v p2 ∨ ¬ p4, where
n = 4. Reading E from left to right we can spell out the
falsifiers of E: p1 = 0, p2 = 0, p3 = "any value", p4 = 1.
Thus if a variable pi is present in E, then we set pi = 0,
and if ¬ pi is present in E, then we set pi = 1. If neither pi
nor ¬ pi is present in E, then we set "any value" which means
that pi can be either 0 or 1.
The set of blockers of E is then the set of complements of
the n-strands that code the falsifiers. Thus, reading E from
left to right we can spell out the blockers of U: "first G",
"then G", "then either G or T", "then T" (recall that 0 is
coded by C and 1 is coded by A).Hence we have 2 blockers here: GGGT and GGTT. Spelling out
the blockers while reading E from left to right may be
considered as giving instructions (for a "robot") for
synthesising the set of blockers for the clause considered.
Hence for E as above the synthesis would go as follows.
- "first G": take a solution R1 with "enough G" (each G
nucleotide is hooked to a solid support at its 3'-end).
- "then G": attach G to all the free 5'-ends of molecules in
R1 getting in this way R2.
- "then either G or T" : divide R2 into two solutions R2,1, R2,2
of equal volume, attach G to all the free 3'-ends of
molecules in R2,1 getting R2,1,G, attach T to all the free 5'-ends
of molecules in R2,2 getting R2,2,T, then mix R2,1,G with
R2,2,T getting R3.
- "then T": attach G to all the free 5'-ends of molecules in
R3 getting in this way R4. Clearly R4 contains "enough" (and
"equal amounts") of all the blockers of U.
If neither p1 nor ¬ p1 is present in a clause, then the
initial mixture R1 will have "enough G" and "enough T" hooked
to a solid support. Then the synthesis proceeds as outlined
above. - (2) An innocent phrase "add B(Φ) to Z1" requires some
computation to ensure that all strands from Z1 to be blocked
will be indeed blocked. This is a part of the laboratory
procedure.
- (3) Since PCR is performed in Step 2, it is clear that our
representation of n-strands and blockers is very simplified.
Clearly, one needs to prime strands to be amplified, and so
all the n-strands will have special prefixes and suffixes
that are needed for a PCR. The blockers are then modified
accordingly.
-
An Algorithm for FIND SAT (and for FULL SAT).
-
Here is an algorithm (
aaaaaa 2) for solving FIND SAT.
ALGORITHM
aaaaaa 2
Input: A Boolean formula Φ of n variables
Steps 1 through 5 are as in
aaaaaa 1.
- 6. Take a sample n-strand s.
- 7. Sequence s.
- 8. Output asg(s).
- 9. Stop.
If the algorithm aaaaaa 1 outputs "YES", then aaaaaa 2
continues in Step 6 by taking a random sample strand from the
solution Z2 which is the "end solution" of aaaaaa 1.-
-
This sample strand is sequenced in Step 7, the resulting
sequence is outputed in Step 8, and aaaaaa 2 stops in Step 9.
It is easy to see that this algorithm aaaaaa 2 either (1)
stops and outputs "NO", or (2) stops and outputs "YES ".
Case (1) holds if and only if Φ is not satisfiable, and case
(2) holds if and only if Φ is satisfiable and is an
assignment satisfying Φ.
-
It should be clear by now, that by iterating PCR we can find
out all the assignments that satisfy Φ.
-
The Full Satisfiability Problem (FULL SAT) is to determine
whether or not an arbitrary Boolean formula Φ is
satisfiable, and if it is, to give all assignments satisfying
Φ.
-
Here is an algorithm
aaaaaa 3 for solving FULL SAT.
ALGORITHM
aaaaaa 3
Input: A Boolean formula Φ of n variables.
Steps 1 through 8 are as in calA
2.
- 9. Add b(s)
- 10. PCR
- 11. PCR Successful?
If so, go to 6.
If not, go to 12. - 12. Output "END"
- 13. Stop.
-
-
In Step 9 we add b(s) blocking in this way strands
representing the last successful assignment asg(s) that we
found. The resulting solution Z3 is then PCR'ed in Step 10
yielding solution Z4; the blocked strands s are then not
multiplied by the PCR.
-
In Step 11 we check whether or not the PCR from Step 10 was
successful. This is the case if the volume of Z4 is "clearly"
larger than the volume of Z3.
-
If this PCR was not successful, then we proceed to Step 12
printing "END", and then the algorithm stops.
-
If this PCR was successful, then we go back to Step 6 and
repeat the cycle of discovering a new assignment satisfying
-
Φ and checking whether there are more assignments that
satisfy Φ.
-
In order to facilitate a better understanding of experimental
implementation of the present invention we give below a
description of a blocking procedure verified by our
laboratory experiments.
PROCEDURE
-
To experimentally test the principle of blocking, a single-stranded
DNA molecule 75 nucleotides in length was
synthesized (ISOGEN Bioscience BV Maarsen, The Netherlands).
This molecule represents a potential solution to a computing
problem and functions as the template molecule, which can be
amplified by PCR. Specific primers (ISOGEN Bioscience BV
Maarsen, The Netherlands) were obtained so that the template
could by multiplied by PCR (fig. 5). The five nucleotides at
the 5' end of the PNA blocking molecules are the same as the
five nucleotides at the 3' end of one of the primers used
(fig. 5). This region of overlap of five nucleotides results
in a situation where either a PNA blocker or a primer can
bind to the template. By hybridizing with their
complementary template sequence the PNA molecules prevent
hybridization of one of the primers with the template.
Because PNA's cannot be extended by DNA polymerases,
hybridization of PNA's results in 'blocking' of the
polymerization.
-
Two different 13-mer PNA blocker-molecules were synthesized
(ISOGEN Bioscience BV, Maarsen, The Netherlands). PNA's were
chosen as blocking molecules because they hybridize sequence-selectively
with DNA and do so with a higher affinity, so at
higher temperature, than comparable DNA molecules. This is
incorporated in the PCR by lowering the temperature after
melting to a temperature at which the PNA blockers can anneal
to the template DNA. After this step the temperature is
lowered further to the annealing temperature of the DNA
primers.
One PNA blocker molecule, called B2, is perfectly
complementary to a region of the template whereas blocker B3
has a mismatch one nucleotide from the 3' terminus. This
mismatch should result in a very small difference in
hybridization stability between B2 and B3, so a decrease in
blocking efficiency of B3 relative to B2. If it would be
possible to detect this decrease in blocking efficiency, one
could use blocking to discriminate between solution molecules
differing from one another by only one nucleotide. The
positive control is the PCR mixture with template DNA but
without PNA blocker. With this mixture normal amplification
by PCR should be observed. As a negative control a PCR
mixture containing all components, except the DNA template,
is taken along. After the PCR the resulting DNA is analyzed
on an ethidiumbromide stained polyacrylamid gel (fig. 5).
-
As can be seen from the gel in figure 5, addition of any of
the two blockers B2 or B3 clearly reduces the amount of
product formed in the PCR. In case of B2, the amount of PCR
product is below the detection limit of the gel. If B3 is
used, a faint band can be seen. When the perfectly
complementary blocker B2 is used, relatively more reduction
is achieved than with the one mismatch blocker B3, so even
one mismatch can indeed yield a detectable reduction of
blocking efficiency. Quantification of the double-stranded
DNA (dsDNA) concentration after PCR (fig. 5) indicates that
addition of B2 reduces the concentration of dsDNA from 2.3
ng/µL to 0.2 ng/µL, a reduction of 91%. Addition of B3 also
results in less dsDNA after PCR, though the reduction of 78%
is significantly less than the 91% achieved by B2.
-
From the results of the described experiment it can be
concluded that blocking of PCR amplification using specific
PNA blockers is possible. Using a fully complementary PNA
blocker, a 91% reduction in the amount of dsDNA after PCR can
be achieved. If, however, a one-mismatch blocker is used, the
blocking efficiency is reduced significantly to 78%. These
results show that the method of blocking by PNA is possible.
It is clear that the percentages of blocking can be further
improved. Further evaluation of the optimal conditions for
blocking could easily be performed using the LightCycler™
(Roche Diagnostics Nederland B.V., Almere, The Netherlands).
-
Another objective was to prove that a significant reduction
in blocking efficiency could be detected if a PNA blocker
with one mismatch was used. If it would be possible to detect
this decrease in blocking efficiency, one could use blocking
to discriminate between DNA molecules differing from one
another by only one nucleotide. For this purpose the blocker
B2, which had a fully complementary sequence to part of the
template was used and compared with B3, which had one
mismatch at the second nucleotide from the 3' terminus.
Theoretically one would expect the association of B2 to the
template to be just somewhat stronger than that of B3. From
the results of the experiment described it can be concluded
that under the experimental conditions tested B2 blocks
better than B3, though the difference is relatively small.
This relatively small decrease in blocking efficiency (13%)
can be explained by the problem of the relatively slow
cooling of the PCR apparatus to annealing temperature, during
which the one-mismatch blocker can anneal to the template.
Another reason is that the mismatch in B3 is one nucleotide
from the 3' terminus. According to recent literature (Igloi,
1998) this results in a very small difference in
hybridization stability between B2 and B3, so only a very
small decrease in blocking efficiency of B3. If the mismatch
had been in the third or fourth position from the 3' terminus
the difference in hybridization stability between B2 and B3
would have been considerably larger, resulting in a bigger
difference in blocking efficiency. So, the comparison between
B2 and B3 is in fact the one at which one would expect to see
a very small difference in blocking efficiency. This
difference could be detected, indicating that the
experimental setup, despite the limitations described above,
is quite effective.
Brief description of the figures
-
Figure 1. Schematic representation of a method in which
detection of a biological molecule of a combinatorial library.
may be blocked by a blocking agent. In this example the
blocker comprises a sequence that is complementary to a part
of the nucleic acid sequence of the biological molecule
thereby allowing hybridization. Both the biological molecule
and the blocker comprise a fluorescent label (indicated with
an open circle). Through the close proximity of the labels of
the blocker and the biological molecule, fluorescence of at
least the label associated with the biological molecule is
quenched (FRET; Fluorescence: Resonance Energy Transfer)
thereby not allowing detection of the biological molecule
that is associated with the blocker. Quenching of course does
not take place when no blocker is associated, thus allowing
detection of the label associated with the biological
molecule when the biological molecule is not associated with
the blocker.
-
Figure 2. Schematic representation of a method in which
detection of a biological molecule of a combinatorial library
may be blocked by a blocking agent. In this example the
blocker comprises a sequence that is complementary to a part
of the nucleic acid sequence of the biological molecule
thereby allowing hybridization. Hybridization of the blocker
prevents at least in part hybridization of a PCR primer
thereby disabling at least in part the amplification of
biological molecule nucleic acid and thereby subsequent
detection. Amplification of biological molecules, not
comprising a blocker, is not impaired thereby allowing
amplification and subsequent detection of unblocked
biological molecules.
-
Figure 3. Schematic representation of a method in which
detection of a biological molecule of a combinatorial library
may be blocked by a blocking agent. In this example the
blocker comprises a protein nucleic acid (PNA) sequence that
is complementary to a part of the nucleic acid sequence of
the biological molecule thereby allowing hybridization.
Hybridization of the blocker in this example, inhibits a
selectable quality such as a restriction site, thereby in the
case of a restriction site, disallowing cloning of the
blocked biological molecule. When the detection system
comprises detection of cloned biological molecules then
inhibition of the selectable quality results in blocking of
detection of blocked biological molecules.
-
Figure 4. Schematic representation of an example wherein the
blocker is associated to a solid support. The solid support
may be a bead as depicted and/or a surface. In this schematic
representation the blocker is associated with a biological
molecule of the library.
-
Figure 5. Photograph of an ethidiumbromide stained 12%
polyacrylamid gel containing separated DNA after PCR. On top
is indicated if template DNA and/or PNA was added to the PCR
mixture. Each PCR mixture also contained 2.5 * 10-7 M of both
primers OMP351 and P1A, 0.25 mM of each dNTP and 5 U SuperTaq
(HT Biotechnologies Ltd., Cambridge, UK) polymerase in a
total volume of 100 µl SuperTaq buffer. Before starting the
PCR 70 µl of mineral oil is put on top to avoid evaporation.
The PCR was performed on a hybaid™ thermal reactor (HYBAID,
Middlesex, UK), using to following program: 10' 95°C, 17
cycles {1' 95°C, 1' 56°C, 1' 48°C, 1' 72°C}, 10' 72°C. Of each
sample 20 µl was mixed with 5 µl loading buffer and separated
on a 12% polyacrylamid gel at 10 V/cm for 2 hours (detailed
procedure: J. Sambrook, E.F. Fritsch, T. Maniatis, Molecular
Clonin: A laboratory manual. Cold Spring harbor Laboratory
press, 1989). DNA was visualized with UV after
ethidiumbromide staining. DNA concentration after PCR was
determined using the PicoGreen dsDNA Quantitation Kit
(Molecular Probes Eugene, Oregon, USA) and a LS50B
luminescence spectrometer with well-plate reader (Perkin-Elmer
Corp., Analytical Instruments, Norwalk CT, USA).
